What exactly is a wave? What are waves, specifically light waves made of? Do points in a wave move along with the wave?
I was reading this post and I have become utterly confused with the concept of a wave. First, is a wave made up of particles or not? (Then again light is said to both a wave and a particle.) I am really confused. Is a light wave a collection of photons?
Second, in that post, one person mentions "material waves". I'm assuming these are waves like water waves and waves on strings/ropes, which are really made of particles. I understand the answer in context of waves in ropes, the atoms are not moving from one point to another. In water waves the water molecules do move from one point to another, don't they?
So what are the other "non-material waves" made of?
 A: A wave is a phenomenon where a change in a field in one place affects the field in their surroundings. For example, when you push a table away from you, you actually push only the table parts which are really near you, thus you compress that part of the table and due to the increase in pressure, the pressure and density of the next part of the table is affected and it is compressed as well. This continues again and again until the compression wave gets to the other side and the last part of the table rarefacts.
When you throw a rock into a pond, you  make the water in one point go lower than its surroundings, but due to surface tension and gravity, the water next to it pulls it up (and it pulls the surrounding water down), which makes a wave in the height of the surface (a water wave).
Notice that in both cases the wave was the traveling of a disturbance in a field which is connected to matter (pressure, height of surface, etc...). A wave is always a disturbance in a field but this field doesn't need to be connected to matter.
For example, light is also a wave: as said by UrasGungorPhys, when you change the electric and magnetic field in one spot, you affect the fields in the places next to it, thus creating an electromagnetic wave. The "particle" nature of light is a whole different subject but tl;dr: light is a wave with quantized energy levels which are in a certain modern way called "particles". Light isn't made of particles as you think for water waves and sound waves, we just call the energy packets of the wave "particles". To understand why we do that there is a lot of quantum mechanics to explain first and it is beyond the scope of this answer :)
A: In waves particles don't have to travel. For sound waves, they oscillate back and forth with pressure changes in the air, or solid/liquid materials. For water waves, water molecules move in orbits, you can tell this because when water waves hit the shore, the water level doesn't instantly change, they don't fill the shore, so the molecules move back.
When you say material waves are 'made of' something, it's more accurate to say material waves travel through space through a material medium. The particles are transferring energy in the form of a wave.
Light waves aren't 'made of' anything, they are a pair of disturbances in electric and magnetic fields, hence their name, electro-magnetic waves. When an electric field changes, it creates a magnetic field, and when a magnetic field changes, it creates an electric field. To understand how this works, you can look into Maxwell's equations of electromagnetism, specifically, the last two.
The wave-particle duality was discovered first for EM waves, with an experimental phenomenon called "The Photoelectric Effect". When light waves at a certain frequency were shined upon a metallic surface, it was observed that electrons popped out of the metal, and unlike water waves carving away rocks at a shore, this happened instantly, indicating that light had a particle property. Yet at this time, the wave properties of light were already well established, eventually leading to the only possible conclusion, light simply had to be both, that it was made of massless particles called photons that carry energy, had momentum, and also had wave properties.
Later though, this was discovered to be true for more than just photons, all particles have wave-particle dualities, this was -I think- first proven for electrons, when they were made to create an interference pattern exactly like waves do. These particle waves are called de Broglie waves, this is one of the most fundamental ideas of quantum mechanics, therefore it is very difficult, perhaps impossible to really understand intuitively. It's a very shocking way the universe works.
So to fully understand the behavior of quantum particles and light waves, both theories are used, in certain situations. Light wave events like diffraction, interference, polarization, etc. are fully describable mathematically by wave theories, other events like the photoelectric effect, Compton scattering, etc. are explained by the particle theories.
A: The word "wave" is an everyday word, and used for seas and lakes since time immemorial.
When mathematics started being used to describe physical phenomena, it was found that the solutions of the wave equation described the phenomenon not only on water but also on strings and sound in general. The simplest solutions are sine and cosine functions, and then, when used in conjunction with mathematical expansions all disturbances in water and sound can be described by wavepackets (as in soliton solutions) of wave equations.
The wave equations in sounds and water describe the collective transfer of energy between matter  in the material displaying the wave. The material is necessary.

Do points in a wave move along with the wave?

There are longitudinal waves, where the particles involved in the energy transfer move back and forth, and transverse where the particles move up and down as the energy is transferred. One has to study the formulas describing specific boundary conditions.
Then came the quantum mechanics revolution. There were puzzles in the data that could not be described with the classical theories of the time, and slowly quantization took hold to describe the micro-world of atoms, molecules, nuclei  and  particle physics .
Light classically is described by a wave equation similar to the water and sound waves, Maxwell's equation. BUT when the quantized nature of the electromagnetic wave is examined, it is experimentally seen that a large number of photons make up classical light, each photon having just $energy=hν$, $spin=+/-1$ and mass zero.


Single-photon camera recording of photons from a double slit illuminated by very weak laser light. Left to right: single frame, superposition of 200, 1’000, and 500’000 frames.

The quantum mechanical wave equation describing the photons is a quantized maxwell equations.
Spin 1/2 praticles obey the Dirac equation and integer spin the Klein Gordon equation,
The waves of these equations are not energy or matter waves, they are probability waves, they predict how probable is it to find a particle at an (x,y,z,t). This explains the double slit one photon at a time linked above, where single photons seem random but their collection shows the interference pattern of a light beam.
So water, air, land , string waves are  energy momentum in motion waves.
Light waves also . BUT photons that make up the light waves are probability waves as all waves in the domain of quantum mechanics.
